In the 1980s, van den Berg speculated that for all parallelepipeds the gap between the first two Dirichlet eigenvalues is bounded below by a constant. Yau subsequently formulated the fundamental gap conjecture:

For all convex domains in $\R^n$, the gap between the first two Dirichlet eigenvalues is bounded below by $\frac{3 \pi^2}{d^2}$, where $d^2$ is the diameter of the domain.

This talk concerns the spectral gap between Dirichlet eigenvalues of convex domains in $\R^n$, and in particular, the fundamental gap of simplices and triangles. I will discuss recent progress with Z. Lu on the fundamental gap conjecture for triangles and simplices, new connections between Neumann eigenvalues and Dirichlet gaps, and demonstrate a relationship between the fundamental gap and Bakry-Emery geometry. In conclusion, I will offer ideas and open problems.

It is well known that the discretization of a finite element method results in a linear system of equations, in which the matrix and the load vector are usually computed by numerical integration. The inexact integration may lead to a different linear system, and consequently, produce a different finite element solution. In this talk, we will first discuss the existing results on the impact of quadrature rules on the finite element approximation in the energy norm. Then, a sharp estimate on the convergence rate of the finite element approximation with numerical integration for linear functionals will be presented. This is the joint work with Ivo Babuska and Uday Banerjee.

Relations in tautological ring of moduli spaces of stable curves can produce universal equations for Gromov-Witten invariants for all compact symplectic manifolds. A typical example of such equations is the WDVV equation, which is a genus-0 equation and gives the associativity of the quantum cohomology. Finding such relations in higher genera is a very difficult problem. I will talk about some topological recursion relations for all genera which was proved in a joint paper with R. Pandharipande. Some of these relations can be used to prove a conjecture of Kefeng Liu and Hao Xu.

We extend some recent results of Lubinsky, Levin, Simon, and Totik
from measures with compact support to spectral measures of
Schr\"odinger operators on the half-line. In particular, we define a
reproducing kernel $S_L$ for Schr\"odinger operators and we use it to
study the fine spacing of eigenvalues in a box of the half-line
Schr\"odinger operator with perturbed periodic potential. We show that
if solutions $u(\xi, x)$ are bounded in $x$ by $e^{\epsilon x}$
uniformly for $\xi$ near the spectrum in an average sense and the
spectral measure is positive and absolutely continuous in a bounded
interval $I$ in the interior of the spectrum with $\xi_0\in I$, then
uniformly in $I$
$$\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)} \rightarrow
\frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},$$ where
$\rho(\xi)d\xi$ is the density of states.
We deduce that the eigenvalues near $\xi_0$ in a large box of size $L$
are spaced asymptotically as $\frac{1}{L\rho}$. We adapt the methods
used to show similar results for orthogonal polynomials.